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The Debt Retirees Owe to Fibonacci

Lenny had a problem. A close friend of his invested some money a few years prior, in a local Italian bank — in Pisa actually — that had promised him steady interest of 4 percent per month. Rather than sitting by and letting the money rapidly grow and compound over time, Lenny’s friend started withdrawing large and irregular sums of money from the account every few months. These sums were soon exceeding the interest he was earning and the whole process was eating heavily into his capital. To make a long story short, Lenny was approached by this friend and asked how long the money would last, if he kept up these withdrawals. Reasonable question, no?

Now, if I were Lenny, I would have pulled out my handy HP business calculator, entered the cash flows, pushed the relevant buttons and quickly obtained the answer. In fact, with any calculator these sorts of questions can be answered quite easily using the technique known as “present value analysis” — which is something all business school professors teach their finance students on the first day of class.

Unfortunately, Lenny didn’t have access to an HP business calculator that performed the necessary compound interest calculations. To be honest, Lenny didn’t have a calculator at all because they hadn’t been invented yet. You see, Lenny was asked this question over 800 years ago, in the early part of the 13th century. But to answer it — which he certainly did — he invented a technique that today is called present value analysis. Yes. The one I mentioned we teach business students.

You might have heard of Lenny by his more formal name: Leonardo Pisano filius (“family,” in Latin) Bonacci, a.k.a. Fibonacci (1170-1250), probably the most famous mathematician of the Middle Ages.

In fact, Fibonacci helped solve his friend’s problem, wrote the first commercial mathematics textbook Liber Abaci and introduced the methodology for solving complicated questions involving interest rates. Let me repeat. His technique is still used and taught with slight refinements to college students 800 years later. Now that is academic immortality! (He published and his name hasn’t perished yet.)

The Spending Rate: A Burning Question

Let’s translate Fibonacci’s mostly hypothetical 800 year-old puzzles into a problem with more recent implications. Imagine that you are thinking about retiring and to date have managed to save a total of $300,000 in your retirement account. For now, I’ll stay away from a discussion about taxes or the account type. Allow me to further assume that you are entitled to a retirement pension income of $25,000 per year. This is the sum total of your Social Security and corporate pension plans — but the $25,000 is not enough. You need at least a total of $55,000 per year, to maintain your current standard of living. This leaves a gap of approximately $30,000 per year, which you hope to fill with your $300,000 nest egg. The pertinent question, then, becomes: Is the $300,000 enough to fill the budget deficit of $30,000 per year? And if not, how long will the money last?

As you probably suspected intuitively by now your $300,000 nest egg is not enough. Think about it this way: The ratio of $30,000 per year (the income you want to generate) divided by the original $300,000 (your nest egg) is 10 percent. There is no financial instrument that I am aware of that can generate a consistent and reliable 10 percent per year. If you don’t want to risk any of your hard earned nest egg in today’s volatile environment, the best you can hope for is about 3 percent after inflation is accounted for, and even that is pushing it. Sure, you might think that you are earning 5 percent guaranteed by a bank, or 5 percent in dividends or 5 percent in bond coupons, but an inflation rate of 2 percent will erode the true return to a mere 3 percent. Needless to say, 3 percent will only generate $9,000 per year in interest from your $300,000 nest egg. That is a far cry from (actually $21,000 short of) the extra $30,000 you wanted to extract from the nest egg.

You have no choice. In retirement you will have to eat into your principal.

In my personal experience talking to retirees and soon-to-be retirees, I find that this realization is one of the most difficult concepts they must come to grips with. Some people simply refuse to spend principal and submit to a reduced standard of living. Principal is sacred and they agree to live on (and adapt to) interest income. But, in today’s low interest-rate environment, let alone once you account for income taxes, living on interest only will eventually lead to a greatly reduced standard of living over time.

So once you accept that actually depleting your nest egg is necessary, the next — and much more relevant — question becomes, If I start depleting capital, how long before there is nothing left?

Here is one of the many problems that Fibonacci posed in his book Liber Abaci, which may seem like just another problem, but is actually the intellectual inspiration for the equation illustrated on the previous page.

“…On Problems of Travellers and Also Similar Problems: A Certain man proceeding to Lucca on business to make a profit doubled his money, and he spent there 12 denari. He then left and went through Florence; he there doubled his money and he spent 12 denari. Then, he returned to Pisa; doubled his money and spent 12 denari, and it is proposed that he had nothing left. It is sought how much he had in the beginning…” (Translation source: L.E. Sigler)

With a little bit of imagination you can translate Fibonacci’s 800 year-old traveler into a modern day retiree who starts retirement with an unknown sum of money. That is the variable to be solved for. The money is invested in a bank account that doubles its value every dozen years, which is an interest rate of approximately 6 percent per year. Now, at the end of each year the retiree withdraws 1 denar (or dollar, euro, peso) from the bank account, and spends it. This growth and spending process continues for three-dozen years (i.e. 36 withdrawals), at which point the money runs out. Fibonacci’s question is: How much money did the retiree — who indeed wants to spend his time traveling — begin with? Remember that this problem was posed 800 years ago, in the year 1200. Retirement challenges might not be as contemporary as you think.

To sum up, Fibonacci’s genius was that he broke-down complicated compound interest calculations — taking place across different periods of time — by bringing the cash flows all back to one focal point in time and manipulating those values on the same date. He eliminated the messy time dimension.

So, how long will your money last in retirement? Leonardo Fibonacci’s answer is as follows. The present value of your retirement withdrawals, from the date of retirement until the unknown date the funds are exhausted, must exactly equal the value of your nest egg. Write it down mathematically, invert the expression and solve for t, and you are left with the expression at the very top of this column. This crucial computation is No. 1 from among what I believe are the 7 most important equations for your retirement.

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